We will deal with the case for \(k=0\) , for the other cases one can follow the same argument by induction. Now we have the following short exact sequence on \(\mathcal {X}_{0}=\mathcal {Y}_{-}\cup _{\mathcal {D}}\mathcal {Y}_{+}\) :
\(0\rightarrow \mathcal {O}_{\mathcal {X}_{0}}\rightarrow \mathcal {O}_{\mathcal {Y}_{-}}\oplus \mathcal {O}_{\mathcal {Y}_{+}}\rightarrow \mathcal {O}_{\mathcal {D}}\rightarrow 0.\) 
Since \(\varphi :\mathcal {O}_{\mathcal {X}_{0}}\rightarrow \mathcal {F}\)  is an admissible orbifold PT pair, tensoring (REF ) with
\(\bar{\mathbf {I}}^\bullet =\lbrace \mathcal {O}_{\mathcal {X}_{0}}\rightarrow \mathcal {F}\rbrace \)  we have the following exact triple of complexes
\(0\rightarrow \bar{\mathbf {I}}^\bullet \rightarrow \bar{\mathbf {I}}^\bullet _{-}\oplus \bar{\mathbf {I}}^\bullet _{+}\rightarrow \bar{\mathbf {I}}^\bullet _{\mathcal {D}}\rightarrow 0\) 
where \(\bar{\mathbf {I}}^\bullet _{\pm }\)  and \(\bar{\mathbf {I}}^\bullet _{\mathcal {D}}\)  denote the restriction of \(\bar{\mathbf {I}}^\bullet \)  on \(\mathcal {Y}_{\pm }\)  and \(\mathcal {D}\)  respectively. By using [[1]}, Corollary 3.9 and Proposition 3.23] for the admissibility of sheaves and using [[2]}, Remark 3.3-(2)] (see also [[3]}, Remark 2.2.6.15]) for the purity of sheaves, \(\varphi _{\pm }:\mathcal {O}_{\mathcal {Y}_{\pm }}\rightarrow \mathcal {F}|_{\mathcal {Y}_{\pm }}\)  are also admissible orbifold PT pairs and \(\varphi _{\mathcal {D}}:\mathcal {O}_{\mathcal {D}}\rightarrow \mathcal {F}|_{\mathcal {D}}\)  is a quotient. By applying \(R\mathrm {Hom}^\bullet (\bar{\mathbf {I}}^\bullet ,\cdot )\)  to (REF ), we have
\(\cdots \rightarrow \mathrm {Ext}^{-1}(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}})\rightarrow \mathrm {Hom}(\bar{\mathbf {I}}^\bullet ,\bar{\mathbf {I}}^\bullet )\rightarrow \mathrm {Hom}(\bar{\mathbf {I}}^\bullet _{-},\bar{\mathbf {I}}^\bullet _{-})\oplus \mathrm {Hom}(\bar{\mathbf {I}}^\bullet _{+},\bar{\mathbf {I}}^\bullet _{+})\rightarrow \mathrm {Hom}(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}})\) 
and
\(\cdots \rightarrow \mathrm {Ext}^{2}(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}})\rightarrow \mathrm {Ext}^3(\bar{\mathbf {I}}^\bullet ,\bar{\mathbf {I}}^\bullet )\rightarrow \mathrm {Ext}^3(\bar{\mathbf {I}}^\bullet _{-},\bar{\mathbf {I}}^\bullet _{-})\oplus \mathrm {Ext}^3(\bar{\mathbf {I}}^\bullet _{+},\bar{\mathbf {I}}^\bullet _{+})\rightarrow \mathrm {Ext}^3(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}}).\) 
Since \(\mathcal {F}\)  is a pure 1-dimensional admissible sheaf and \(\mathrm {coker}\,\varphi \)  is normal to \(\mathcal {D}\) , the complex \(\bar{\mathbf {I}}^\bullet _{\mathcal {D}}\)  is quasi-isomorphic to an ideal sheaf of some points in \(\mathcal {D}\) . Then we have \(\mathrm {Ext}^{-1}(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}})=0\)  and \(\mathrm {Ext}^{2}(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}})_{0}=0\) , where the vanishing of the latter is proved by using the similar argument in [[4]}, Lemmas 5.15 and 5.16] to show \(\mathrm {Hom}(\bar{\mathbf {I}}_{\mathcal {D}}^\bullet ,\bar{\mathbf {I}}^\bullet _{\mathcal {D}})\cong \mathcal {O}_{\mathcal {D}}\)  and then tensoring with \(\omega _{\mathcal {D}}\)  to apply Serre duality in [[4]}, Lemma 5.11] since \(\mathcal {D}\)  is a smooth projective Deligne-Mumford stack of dimension 2 by assumption. By [[4]}, Lemma 5.16], we have \(\mathrm {Hom}(\bar{\mathbf {I}}^\bullet _{\pm },\bar{\mathbf {I}}^\bullet _{\pm })_{0}=\mathrm {Ext}^3(\bar{\mathbf {I}}^\bullet _{\pm },\bar{\mathbf {I}}^\bullet _{\pm })_{0}=0\)  which implies \(\mathrm {Hom}(\bar{\mathbf {I}}^\bullet ,\bar{\mathbf {I}}^\bullet )_{0}=\mathrm {Ext}^3(\bar{\mathbf {I}}^\bullet ,\bar{\mathbf {I}}^\bullet )_{0}=0\)  by taking the traceless part of (REF ) and (REF ) respectively. Applying the similar argument for the cases in higher degrees and lower degrees, the proof is completed.
